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We show that $E_2(n)\\geq \\Omega(n\\sqrt{\\log n})$, which answers a question of Eisenbrand, Pach, Rothvo\\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that $\\lfloor\\frac{1}{3}n^2\\rfloor\\leq E_3(n)\\leq \\frac{3}{8}n^2+O(n^{3/2})$.\n  Let $W_d(n)$ be the maximum number of pairwise nonparallel unit distance pairs i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1007.2775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-07-16T13:45:56Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"05b36a983115f2c9dcbca0e05e74f729940f082718da85d4bc14325efddcc9d3","abstract_canon_sha256":"8a8777f9297b2c500c3e99e61e09a155d90e4030783b816887b075c7e2b73abb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:46.170080Z","signature_b64":"vWgc9gFP1zrRZ1Zj7eVae9h0324nYnTQSx7pBsGToWLSKap6XdfZxbtSAz91hGA8fxEPxMaLJVQtIRIVbXDlDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"23dc40d7261c28661fdec7ca07733c30acfbaac4b1b02f83039dca019df1e2c7","last_reissued_at":"2026-05-18T04:14:46.169468Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:46.169468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large convexly independent subsets of Minkowski sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Konrad J. Swanepoel, Pavel Valtr","submitted_at":"2010-07-16T13:45:56Z","abstract_excerpt":"Let $E_d(n)$ be the maximum number of pairs that can be selected from a set of $n$ points in $R^d$ such that the midpoints of these pairs are convexly independent. We show that $E_2(n)\\geq \\Omega(n\\sqrt{\\log n})$, which answers a question of Eisenbrand, Pach, Rothvo\\ss, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that $\\lfloor\\frac{1}{3}n^2\\rfloor\\leq E_3(n)\\leq \\frac{3}{8}n^2+O(n^{3/2})$.\n  Let $W_d(n)$ be the maximum number of pairwise nonparallel unit distance pairs i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2775","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1007.2775","created_at":"2026-05-18T04:14:46.169546+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.2775v1","created_at":"2026-05-18T04:14:46.169546+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2775","created_at":"2026-05-18T04:14:46.169546+00:00"},{"alias_kind":"pith_short_12","alias_value":"EPOEBVZGDQUG","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_16","alias_value":"EPOEBVZGDQUGMH66","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_8","alias_value":"EPOEBVZG","created_at":"2026-05-18T12:26:06.534383+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC","json":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC.json","graph_json":"https://pith.science/api/pith-number/EPOEBVZGDQUGMH66Y7FAO4Z4GC/graph.json","events_json":"https://pith.science/api/pith-number/EPOEBVZGDQUGMH66Y7FAO4Z4GC/events.json","paper":"https://pith.science/paper/EPOEBVZG"},"agent_actions":{"view_html":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC","download_json":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC.json","view_paper":"https://pith.science/paper/EPOEBVZG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.2775&json=true","fetch_graph":"https://pith.science/api/pith-number/EPOEBVZGDQUGMH66Y7FAO4Z4GC/graph.json","fetch_events":"https://pith.science/api/pith-number/EPOEBVZGDQUGMH66Y7FAO4Z4GC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC/action/storage_attestation","attest_author":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC/action/author_attestation","sign_citation":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC/action/citation_signature","submit_replication":"https://pith.science/pith/EPOEBVZGDQUGMH66Y7FAO4Z4GC/action/replication_record"}},"created_at":"2026-05-18T04:14:46.169546+00:00","updated_at":"2026-05-18T04:14:46.169546+00:00"}